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3 is an integer. $\frac 13$ is a rational number. $\sqrt 3$ is an irrational number. But what about $\pi$? Or some other irrational infinitely repeating number that cannot be written as a ratio of square roots and rational numbers?

I know pi can be written compactly as $\frac c{2r}$ or something similar but that's not what I'm after. I'm interested in the numbers that are irrational but not square roots.

Is there a name for that set?

Tan
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Merkava120
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    The numbers you describe could be transcendental numbers. – fwd Jan 22 '22 at 18:17
  • So something like non-algebraic number but limited to square roots and not polynomial roots? I guess it's not quite a quadratic irrational number either, or is it? – GrapefruitIsAwesome Jan 22 '22 at 18:19
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    How is this a [tag:set-theory] question? – José Carlos Santos Jan 22 '22 at 18:20
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    What exactly does it mean to be "written with square roots"? For instance, is $\frac{\sqrt{1 + \sqrt{3}} - \sqrt{5}}{\sqrt{7}-\sqrt{11}}$ "written with square roots"? – Lee Mosher Jan 22 '22 at 18:24
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    from the tradition of compass and straightedge constructions, real numbers that can be expressed using just rational numbers, field operations, and square roots (of positive elements) , are sometimes called the constructible numbers. – Will Jagy Jan 22 '22 at 18:25
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    @fwd Not true, algebraic numbers cannot be written using radicals in general either. – Peter Jan 22 '22 at 18:39
  • @WillJagy This comes close to what is intented. – Peter Jan 22 '22 at 18:40
  • @Peter yes. For the difficult example of $2\cos \frac{2 \pi}{17}, $ see the degree 8 polynomial on page 18 of https://www.google.com/books/edition/Tafeln_complexer_Primzahlen/wt7lgfeYqMQC?q=reuschle++tafeln+complexer+primzahlen&gbpv=1#f=false Actually expressing it ( a root) in square roots is in some Galois Theory books, quite messy – Will Jagy Jan 22 '22 at 18:46
  • @Peter I only meant that a transcendental number fits the OP's description. – fwd Jan 22 '22 at 19:21
  • @Will Jagy: Actually expressing it ( a root) in square roots is in some Galois Theory books, quite messy -- I give an extremely detailed derivation of such an expression on pp. 18-26 of this manuscript (click on "File available"). In fact, only "school algebra" methods are used, and no computational aids are needed to follow the steps. In fact, even paper-and-pencil calculations do not have to be done, except in perhaps one case where I had to make use of $(34)(17) = 578.$ (continued) – Dave L. Renfro Jan 22 '22 at 19:36
  • I know the MS Word format (the .pdf file was generated from) suggests amateurish work, which technically is true since I didn't get any money from doing this :), but there is a reason I didn't prepare this in LaTeX at the time. I may at some later time prepare an expanded LaTeX version, I don't know. (I was asked a few years ago by a fairly well known MAA officer to consider writing an expanded/revised version for one of their book publication series, but thus far I've been too busy with other things.) – Dave L. Renfro Jan 22 '22 at 19:45
  • "even paper-and-pencil calculations do not have to be done" -- I mean by someone reading through the details I've written. Obviously a huge amount of paper-and-pencil calculations were done when I was working out the details! – Dave L. Renfro Jan 22 '22 at 19:50
  • @DaveL.Renfro looks nice. – Will Jagy Jan 22 '22 at 20:31

1 Answers1

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There is a set of numbers called the “constructible numbers” which are the numbers you can get starting from $1$ using addition, subtraction, multiplication, division by a nonzero number, and taking the square root of a nonnegative number. So you would be looking for the non-constructible numbers.

The term “constructible” comes from Ancient Greek geometry. In Greek geometry, you constructed figures by drawing lines (with a straight edge) and circles (with a compass), but no other operations were allowed. The constructible numbers are exactly the lengths that you can construct using a straight edge and compass and starting with a line segment of length $1$. Two of the classic problems of Greek geometry are: is $\pi$ constructible (a problem known as “squaring the circle” - if $\pi$ were constructible, then given a circle, you could make a square with the same area), and is $\sqrt[3]{2}$ constructible (known as “doubling the cube” - if you could construct $\sqrt[3]{2}$, then you could take one cube and make another with twice the volume)? It turns out that both of these numbers are non-constructible.

Mark Saving
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